Deriving identities for Wigner {nj}-symbols

TL;DR

This paper introduces a graphical notation technique to derive identities for Wigner {nj}-symbols, including the Biedenharn-Elliot identity for 6j-symbols and a new identity for the 15j-symbol in 4D spin-foam gravity.

Contribution

It presents a novel graphical method for deriving identities of Wigner {nj}-symbols, extending to new identities relevant for 4D spin-foam models.

Findings

Derived the Biedenharn-Elliot identity using graphical notation

Proposed a new identity for the 15j-symbol in 4D spin-foam gravity

Demonstrated the method's applicability to complex quantum gravity symbols

Abstract

We show how a simple and elegant graphical notation can be used to derive the Biedenharn-Elliot identity for the 6j-symbol and we demonstrate how the same technique can be applied to obtain new identities for the 6j. We then employ the same method also in the context of 4D spin-foam gravity and propose an analogous identity for the 15j-symbol.